Left Termination of the query pattern append3_in_4(g, g, g, a) w.r.t. the given Prolog program could successfully be proven:



↳ Prolog
  ↳ PrologToPiTRSProof

Clauses:

append([], L, L).
append(.(H, L1), L2, .(H, L3)) :- append(L1, L2, L3).
append3(A, B, C, D) :- ','(append(A, B, E), append(E, C, D)).

Queries:

append3(g,g,g,a).

We use the technique of [30].Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

append3_in(A, B, C, D) → U2(A, B, C, D, append_in(A, B, E))
append_in(.(H, L1), L2, .(H, L3)) → U1(H, L1, L2, L3, append_in(L1, L2, L3))
append_in([], L, L) → append_out([], L, L)
U1(H, L1, L2, L3, append_out(L1, L2, L3)) → append_out(.(H, L1), L2, .(H, L3))
U2(A, B, C, D, append_out(A, B, E)) → U3(A, B, C, D, append_in(E, C, D))
U3(A, B, C, D, append_out(E, C, D)) → append3_out(A, B, C, D)

The argument filtering Pi contains the following mapping:
append3_in(x1, x2, x3, x4)  =  append3_in(x1, x2, x3)
U2(x1, x2, x3, x4, x5)  =  U2(x3, x5)
append_in(x1, x2, x3)  =  append_in(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5)  =  U1(x1, x5)
[]  =  []
append_out(x1, x2, x3)  =  append_out(x3)
U3(x1, x2, x3, x4, x5)  =  U3(x5)
append3_out(x1, x2, x3, x4)  =  append3_out(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

append3_in(A, B, C, D) → U2(A, B, C, D, append_in(A, B, E))
append_in(.(H, L1), L2, .(H, L3)) → U1(H, L1, L2, L3, append_in(L1, L2, L3))
append_in([], L, L) → append_out([], L, L)
U1(H, L1, L2, L3, append_out(L1, L2, L3)) → append_out(.(H, L1), L2, .(H, L3))
U2(A, B, C, D, append_out(A, B, E)) → U3(A, B, C, D, append_in(E, C, D))
U3(A, B, C, D, append_out(E, C, D)) → append3_out(A, B, C, D)

The argument filtering Pi contains the following mapping:
append3_in(x1, x2, x3, x4)  =  append3_in(x1, x2, x3)
U2(x1, x2, x3, x4, x5)  =  U2(x3, x5)
append_in(x1, x2, x3)  =  append_in(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5)  =  U1(x1, x5)
[]  =  []
append_out(x1, x2, x3)  =  append_out(x3)
U3(x1, x2, x3, x4, x5)  =  U3(x5)
append3_out(x1, x2, x3, x4)  =  append3_out(x4)


Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

APPEND3_IN(A, B, C, D) → U21(A, B, C, D, append_in(A, B, E))
APPEND3_IN(A, B, C, D) → APPEND_IN(A, B, E)
APPEND_IN(.(H, L1), L2, .(H, L3)) → U11(H, L1, L2, L3, append_in(L1, L2, L3))
APPEND_IN(.(H, L1), L2, .(H, L3)) → APPEND_IN(L1, L2, L3)
U21(A, B, C, D, append_out(A, B, E)) → U31(A, B, C, D, append_in(E, C, D))
U21(A, B, C, D, append_out(A, B, E)) → APPEND_IN(E, C, D)

The TRS R consists of the following rules:

append3_in(A, B, C, D) → U2(A, B, C, D, append_in(A, B, E))
append_in(.(H, L1), L2, .(H, L3)) → U1(H, L1, L2, L3, append_in(L1, L2, L3))
append_in([], L, L) → append_out([], L, L)
U1(H, L1, L2, L3, append_out(L1, L2, L3)) → append_out(.(H, L1), L2, .(H, L3))
U2(A, B, C, D, append_out(A, B, E)) → U3(A, B, C, D, append_in(E, C, D))
U3(A, B, C, D, append_out(E, C, D)) → append3_out(A, B, C, D)

The argument filtering Pi contains the following mapping:
append3_in(x1, x2, x3, x4)  =  append3_in(x1, x2, x3)
U2(x1, x2, x3, x4, x5)  =  U2(x3, x5)
append_in(x1, x2, x3)  =  append_in(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5)  =  U1(x1, x5)
[]  =  []
append_out(x1, x2, x3)  =  append_out(x3)
U3(x1, x2, x3, x4, x5)  =  U3(x5)
append3_out(x1, x2, x3, x4)  =  append3_out(x4)
U31(x1, x2, x3, x4, x5)  =  U31(x5)
APPEND_IN(x1, x2, x3)  =  APPEND_IN(x1, x2)
APPEND3_IN(x1, x2, x3, x4)  =  APPEND3_IN(x1, x2, x3)
U21(x1, x2, x3, x4, x5)  =  U21(x3, x5)
U11(x1, x2, x3, x4, x5)  =  U11(x1, x5)

We have to consider all (P,R,Pi)-chains

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof

Pi DP problem:
The TRS P consists of the following rules:

APPEND3_IN(A, B, C, D) → U21(A, B, C, D, append_in(A, B, E))
APPEND3_IN(A, B, C, D) → APPEND_IN(A, B, E)
APPEND_IN(.(H, L1), L2, .(H, L3)) → U11(H, L1, L2, L3, append_in(L1, L2, L3))
APPEND_IN(.(H, L1), L2, .(H, L3)) → APPEND_IN(L1, L2, L3)
U21(A, B, C, D, append_out(A, B, E)) → U31(A, B, C, D, append_in(E, C, D))
U21(A, B, C, D, append_out(A, B, E)) → APPEND_IN(E, C, D)

The TRS R consists of the following rules:

append3_in(A, B, C, D) → U2(A, B, C, D, append_in(A, B, E))
append_in(.(H, L1), L2, .(H, L3)) → U1(H, L1, L2, L3, append_in(L1, L2, L3))
append_in([], L, L) → append_out([], L, L)
U1(H, L1, L2, L3, append_out(L1, L2, L3)) → append_out(.(H, L1), L2, .(H, L3))
U2(A, B, C, D, append_out(A, B, E)) → U3(A, B, C, D, append_in(E, C, D))
U3(A, B, C, D, append_out(E, C, D)) → append3_out(A, B, C, D)

The argument filtering Pi contains the following mapping:
append3_in(x1, x2, x3, x4)  =  append3_in(x1, x2, x3)
U2(x1, x2, x3, x4, x5)  =  U2(x3, x5)
append_in(x1, x2, x3)  =  append_in(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5)  =  U1(x1, x5)
[]  =  []
append_out(x1, x2, x3)  =  append_out(x3)
U3(x1, x2, x3, x4, x5)  =  U3(x5)
append3_out(x1, x2, x3, x4)  =  append3_out(x4)
U31(x1, x2, x3, x4, x5)  =  U31(x5)
APPEND_IN(x1, x2, x3)  =  APPEND_IN(x1, x2)
APPEND3_IN(x1, x2, x3, x4)  =  APPEND3_IN(x1, x2, x3)
U21(x1, x2, x3, x4, x5)  =  U21(x3, x5)
U11(x1, x2, x3, x4, x5)  =  U11(x1, x5)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 1 SCC with 5 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof

Pi DP problem:
The TRS P consists of the following rules:

APPEND_IN(.(H, L1), L2, .(H, L3)) → APPEND_IN(L1, L2, L3)

The TRS R consists of the following rules:

append3_in(A, B, C, D) → U2(A, B, C, D, append_in(A, B, E))
append_in(.(H, L1), L2, .(H, L3)) → U1(H, L1, L2, L3, append_in(L1, L2, L3))
append_in([], L, L) → append_out([], L, L)
U1(H, L1, L2, L3, append_out(L1, L2, L3)) → append_out(.(H, L1), L2, .(H, L3))
U2(A, B, C, D, append_out(A, B, E)) → U3(A, B, C, D, append_in(E, C, D))
U3(A, B, C, D, append_out(E, C, D)) → append3_out(A, B, C, D)

The argument filtering Pi contains the following mapping:
append3_in(x1, x2, x3, x4)  =  append3_in(x1, x2, x3)
U2(x1, x2, x3, x4, x5)  =  U2(x3, x5)
append_in(x1, x2, x3)  =  append_in(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5)  =  U1(x1, x5)
[]  =  []
append_out(x1, x2, x3)  =  append_out(x3)
U3(x1, x2, x3, x4, x5)  =  U3(x5)
append3_out(x1, x2, x3, x4)  =  append3_out(x4)
APPEND_IN(x1, x2, x3)  =  APPEND_IN(x1, x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof

Pi DP problem:
The TRS P consists of the following rules:

APPEND_IN(.(H, L1), L2, .(H, L3)) → APPEND_IN(L1, L2, L3)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND_IN(x1, x2, x3)  =  APPEND_IN(x1, x2)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof
                    ↳ QDP
                      ↳ QDPSizeChangeProof

Q DP problem:
The TRS P consists of the following rules:

APPEND_IN(.(H, L1), L2) → APPEND_IN(L1, L2)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: